Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420420 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60456 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1657 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,19)(8,23)(9,15,27,33)(10,17)(11,16)(12,18)(13,21)(14,22)(20,30,35,31)(24,29,25,36)(26,34,28,32);(1,2)(3,4)(5,7)(6,13)(8,10)(9,25)(11,21)(12,22)(14,16)(15,31)(17,19)(18,23)(20,26)(24,27)(28,35)(29,32)(30,33)(34,36);(2,4)(5,8,11,22,12,21,10,7)(6,14,17,23,18,19,16,13)(9,24,28,35,27,25,26,20)(15,30,34,36,33,31,32,29);(5,9,10,26,12,27,11,28)(6,15,16,32,18,33,17,34)(7,20,21,25,22,35,8,24)(13,29,19,31,23,36,14,30)Polycyclic presentation:
2^9|a^2=ef,c^2=ei,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327721);